(a) To find the probability that a randomly selected time interval between eruptions is longer than 115 minutes, we need to calculate the area under the normal distribution curve to the right of 115 minutes. This can be done using the z-score.
First, we need to standardize the value of 115 minutes using the formula:
z = (x - μ) / σ
Here, x is the value we want to standardize (115 minutes), μ is the mean (100 minutes), and σ is the standard deviation (35 minutes).
z = (115 - 100) / 35
z ≈ 0.4286
Now, we can look up the area to the right of the z-score 0.4286 in the standard normal distribution table or use a calculator. The area to the right of 0.4286 is approximately 0.3340.
Therefore, the probability that a randomly selected time interval between eruptions is longer than 115 minutes is approximately 0.3340 (rounded to four decimal places).
(b) To find the probability that a random sample of 12 time intervals between eruptions has a mean longer than 115 minutes, we can use the central limit theorem. According to the central limit theorem, when sample size is sufficiently large, the distribution of the sample means approximates a normal distribution.
Since the sample size is 12 and the population standard deviation is known, we can calculate the standard error of the mean (SEM) using the formula:
SEM = σ / √n
Here, σ is the population standard deviation (35 minutes) and n is the sample size (12).
SEM = 35 / √12
SEM ≈ 10.098
Next, we need to standardize the value of 115 minutes using the formula:
z = (x - μ) / SEM
Here, x is the value we want to standardize (115 minutes), μ is the population mean (100 minutes), and SEM is the standard error of the mean (10.098).
z = (115 - 100) / 10.098
z ≈ 1.483
Now, we can look up the area to the right of the z-score 1.483 in the standard normal distribution table or use a calculator. The area to the right of 1.483 is approximately 0.0694.
Therefore, the probability that a random sample of 12 time intervals between eruptions has a mean longer than 115 minutes is approximately 0.0694 (rounded to four decimal places).
(c) Similarly, to find the probability that a random sample of 34 time intervals between eruptions has a mean longer than 115 minutes, we can follow the same steps as in part (b). Calculate the SEM using the formula:
SEM = σ / √n
Here, σ is the population standard deviation (35 minutes) and n is the sample size (34).
SEM = 35 / √34
SEM ≈ 5.988
Standardize the value of 115 minutes:
z = (115 - 100) / 5.988
z ≈ 2.505
Look up the area to the right of the z-score 2.505 in the standard normal distribution table or use a calculator. The area to the right of 2.505 is approximately 0.0061.
Therefore, the probability that a random sample of 34 time intervals between eruptions has a mean longer than 115 minutes is approximately 0.0061 (rounded to four decimal places).
(d) Increasing the sample size has the effect of reducing the standard error of the mean (SEM). As the sample size increases, the SEM decreases, which means the sample mean